However, when starting from the TFM expressions of \(\mathbf {\Gamma }\)  and \(\mathbf {\Phi }\) , the simplest way of obtaining stabilizable and detectable realizations for \(e_i^\top \begin{bmatrix}\mathbf {\Phi }&\mathbf {\Gamma }\end{bmatrix}\)  is to begin by expressing the observable canonical forms (see Chapter 2.1 of [1]}) of these TFMs. As each \(e_i^\top \begin{bmatrix}\mathbf {\Phi }&\mathbf {\Gamma }\end{bmatrix}\)  has a single output, these forms will produce observable realizations of very low order. Staircase-like algorithms [2]} can be employed on these low-order realizations to extract the part that is also controllable.

